Year 1
Week Content
Unit 1: Number (9 weeks)
1
Introduction to
course
2
Straight Lines
2
Sets and Venn
Diagrams
Surds and
Exponents
3
4
Equations
Syllabus Reference
Book/
Chapter
SL 1.6 Approximation: decimal places, significant figures. Students should be able to choose an appropriate degree of accuracy
based on given data. Upper and lower bounds of rounded numbers. If x = 4.1 to one decimal place, 4.05 ≤ x < 4.15. Percentage
errors. Students should be aware of, and able to calculate, measurement errors (such as rounding errors or measurement
limitations). For example finding the maximum percentage error in the area of a circle if the radius measured is 2.5 cm to one
decimal place. Estimation. Students should be able to recognize whether the results of calculations are reasonable. For example
lengths cannot be negative.
Recap of prior knowledge: Solution of linear equations, Solving systems of linear equations in two variables, Graphing linear and
quadratic functions using technology.
SL 2.1 Different forms of the equation of a straight line. Gradient; intercepts. Lines with gradients m1 and m2. Parallel lines m1 =
m2. Perpendicular lines m1 × m2 = − 1. y = mx + c (gradient-intercept form). ax + by + d = 0 (general form). y − y1 = m(x − x1) (pointgradient form). Calculate gradients of inclines such as mountain roads, bridges, etc.
SL 3.5 Equations of perpendicular bisectors. Given either two points, or the equation of a line
segment and its midpoint.
SL 1.8 Use technology to solve: Systems of linear equations in up to 3 variables, Polynomial equations. In examinations, no specific
method of solution will be required. In examinations, there will always be a unique solution to a system of equations. Standard
terminology, such as zeros or roots, should be taught.
SL 2.5 Linear models f(x) = mx+c
Recap of prior knowledge: Number systems: natural numbers ℕ; integers, ℤ; rationals, ℚ, and irrationals; real numbers, ℝ,
Concept and basic notation of sets. Operations on sets: union and intersection. Venn diagrams for sorting data.
Recap of prior learning: Simplification of simple expressions involving roots (surds or radicals), Rationalising the denominator,
Expression of numbers in the form a × 10k, 1 ≤ a < 10, k ∈ ℤ, Evaluating exponential expressions with simple positive exponents
and rational exponents
SL 1.1 Operations with numbers in the form a × 10^k where 1 ≤ a < 10 and k is an integer.
SL 1.5 Laws of exponents with integer exponents.
AHL 1.10 Simplifying expressions, both numerically and algebraically, involving rational exponents.
Recap of prior learning: Solution of linear equations and inequalities, Solution of quadratic equations and inequalities with
rational coefficients, graphing linear functions using technology
SL 2.4 Determine key features of graphs. Maximum and minimum values; intercepts; symmetry; vertex; zeros of functions or
roots of
equations; vertical and horizontal asymptotes using graphing technology. Finding the points of intersection of two curves or lines
using technology.
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1/4
5
Sequences and
Series
SL 1.2 Arithmetic sequences. Use of the formula for the nth term. Applications of arithmetic sequences. Examples include simple
interest over a number of years. Analysis, interpretation and prediction of arithmetic sequences where a model is not perfectly
arithmetic in real life.
SL 1.3 Geometric sequences. Use of the formula for the nth term. Applications. Examples include the spread of disease, salary
increase and decrease and population growth.
SL 1.4 Financial applications of geometric sequences and series: compound interest, annual depreciation. Examination questions
may require the use of technology, including built-in financial packages. Calculate the real value of an investment with an interest
rate and an inflation rate (interest rate – inflation rate). In examinations, questions that ask students to derive the formula will
not be set. Compound interest can be calculated yearly, half-yearly, quarterly or monthly.
SL 1.7 Amortization and annuities using technology. In examinations the payments will be made at the end of the period.
Knowledge of the annuity formula will enhance understanding but will not be examined.
6
Sequences and
SL 1.2 Arithmetic series. Use of sigma notation for sums of arithmetic sequences. The sum of the first n terms of the sequence.
Series
Applications. Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.
SL 1.3 Geometric series. The sum of the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences.
Applications: examples include the spread of disease, salary increase and population growth
AHL 1.11 The sum of infinite geometric sequences.
7
Recap, review and assessment
Unit 2: Functions (8 weeks)
8
Quadratic
Recap of prior learning: Solution of quadratic equations and inequalities with rational coefficients, Graphing quadratic functions
Functions
using technology
SL 2.5 Quadratic models. f(x) = ax2 + bx + c ; a ≠ 0. Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y -axis.
Technology can be used to find roots.
9
Functions
SL 2.2 Concept of a function, domain, range and graph. Function notation, for example f(x) , v(t) , C(n). The concept of a function
as a mathematical model. Example: f(x) = 2 − x, the domain is x ≤ 2, range is f(x) ≥ 0. A graph is helpful in visualizing the range.
Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line y
= x, and the notation f−1(x). Example: Solving f(x) = 10 is equivalent to finding f−1(10). Students should be aware that inverse
functions exist for one to one functions; the domain of f−1(x) is equal to the range of f(x).
SL 2.3 The graph of a function; its equation y = f(x). Students should be aware of the difference between the command terms
“draw” and “sketch”. Creating a sketch from information given or a context, including transferring a graph from screen to paper.
Using technology to graph functions including their sums and differences. All axes and key features should be labelled.
AHL 2.7 Composite functions in context. The notation ( f ∘ g)(x) = f (g(x)). Inverse function f−1, including domain restriction. Finding
an inverse function.
10
Transformations
AHL 2.8 Transformations of graphs. Students will be expected to be able to perform transformations on all functions from the SL
of Functions
and AHL section of this topic, and others in the context of modelling real-life situations. Translations: y = f (x) + b ;y = f (x − a).
Translation by the vector ( 3, −2) denotes horizontal translation of 3 units to the right, and vertical translation of 2 units down.
Reflections: in the x axis y = − f (x), and in the y axis y = f ( − x). Vertical stretch with scale factor p: y = p f (x). Horizontal stretch
with scale factor 1/q : y = f (qx). x and y axes are invariant. Composite transformations. Students should be made aware of the
significance of the order of transformations. Example: y = x2 used to obtain y = 3x2 + 2 by a vertical stretch of scale factor 3
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and 2/4
1/5
1/14
1/15
1/16
11
Exponentials
12
Logarithms
followed by a translation of (0, 2). Example: y = sinx used to obtain y = 4sin2x by a vertical stretch of scale factor 4 and a
horizontal stretch of scale factor 12.
SL 2.5 Exponential growth and decay models. f (x) = kax + c, f (x) = ka−x + c (for a > 0), f(x) = kerx + c Equation of a horizontal
asymptote.
AHL 2.9 Exponential models to calculate half-life. Logistic models: f(x) = L/(1 + Ce−kx) ; L, C, k > 0. The logistic function is used in
situations where there is a restriction on the growth. For example population on an island, bacteria in a petri dish or the increase
in height of a person or seedling. Horizontal asymptote at f (x) = L is often referred to as the carrying capacity.
SL 1.5 Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.
𝑥
AHL 1.9 Laws of logarithms: logaxy = logax + logay / loga = logax – logay / logaxm = mlogax for a, x, y > 0. In examinations, a will
2/1
2/2
𝑦
13
Approximations
and Error
14
Modelling
15
Direct and Inverse
Variation
Bivariate Statistics
16
equal 10 or e.
AHL 2.9 Natural logarithmic models: f (x) = a + blnx
AHL 2.10 Scaling very large or small numbers using logarithms. Linearizing data using logarithms to determine if the data has an
exponential or a power relationship using best-fit straight lines to determine parameters. Choosing a manageable scale, for
example for data with a wide range of values in one, or both variables and/or where the emphasis of a graph is the rate of
growth, rather than the absolute value. Interpretation of log-log and semi-log graphs. In examinations, students will not be
expected to draw or sketch these graphs. Interpretation of log-log and semi-log graphs.
SL 2.5 Modelling with the following functions: Linear models. f(x) = mx + c. Including piecewise linear models, for example
horizontal distances of an object to a wall, depth of a swimming pool, mobile phone charges. Cubic models: f(x) = ax3 + bx2 + cx +
d.
SL 2.6 Modelling skills: Use the modelling process described in the “mathematical modelling” section to create, fit and use the
theoretical models in section SL 2.5 and their graphs. Develop and fit the model: Given a context recognize and choose an
appropriate model and possible parameters. Determine a reasonable domain for a model. Find the parameters of a model. By
setting up and solving equations simultaneously (using technology), by consideration of initial conditions or by substitution of
points into a given function. Test and reflect upon the model: Comment on the appropriateness and reasonableness of a model.
Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the
situation. Use the model: Reading, interpreting and making predictions based on the model. Students should be aware of the
dangers of extrapolation.
AHL 2.9 Piecewise models. In some cases, parameters may need to be found that ensure continuity of the function, for example
find a to make f (x) = { 1 + x , 0 ≤ x < 2; ax2 + x , x ≥ 2 continuous. The formal definition of continuity is not required. In
examinations, students may be expected to interpret and use other models that are introduced in the question.
Recap of prior learning: power functions
SL 2.5 Direct/inverse variation: f(x) = axn, n ∈ ℤ. The y-axis as a vertical asymptote when n < 0.
SL 4.4 Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient, r. Technology should be used to
calculate r. However, hand calculations of r may enhance understanding. Critical values of r will be given where appropriate.
Students should be aware that Pearson’s product moment correlation coefficient (r) is only meaningful
for linear relationships. Scatter diagrams; lines of best fit, by eye, passing through the mean point. Positive, zero, negative; strong,
weak, no correlation. Students should be able to make the distinction between correlation and causation and know that
correlation does not imply causation. Equation of the regression line of y on x. Use of the equation of the regression line for
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2/5
1/15D
and 2/6
2/7
17
prediction purposes. Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b. Technology should be
used to find the equation. Students should be aware of the dangers of extrapolation and that they cannot always reliably make a
prediction of x from a value of y, when using a y on x line.
SL 4.10 Spearman’s rank correlation coefficient, rs. In examinations Spearman’s rank correlation coefficient, rs, should be found
using technology. If data items are equal, ranks should be averaged. Awareness of the appropriateness and limitations of
Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient, and the effect of outliers on each.
Students should be aware that Pearson’s product moment correlation coefficient is useful when testing for only linearity and
Spearman’s correlation coefficient for any monotonic relationship. Spearman’s correlation coefficient is less sensitive to outliers
than Pearson’s product moment correlation coefficient.
AHL 4.13 Regression with non-linear functions. Evaluation of least squares regression curves using technology. In examinations,
questions may be asked on linear, quadratic, cubic, exponential, power and sine functions. Sum of square residuals (SSres) as a
measure of fit for a model. The coefficient of determination (R2). Evaluation of R2 using technology. R2 gives the proportion of
variability in the second variable accounted for by the chosen model. Awareness that R2 = 1 − SSres/SStot
and hence = 1 if SSres = 0, may enhance understanding but will not be examined. Awareness that many factors affect the validity
of a model and the coefficient of determination, by itself, is not a good way to decide between different models. The connection
between the coefficient of determination and the Pearson’s product moment correlation coefficient for linear models.
Further depth into topics addressed in SL 2.6 and AHL 2.9
Non-Linear
Modelling
18
Recap, review and assessment
Unit 3: Probability and Statistics (10 weeks)
19
Core topics:
SL 4.5 Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event. The probability of an
Probability
event A is P(A) = n(A)/n(U). The complementary events A and A′ (not A). Sample spaces can be represented in many ways, for
example as a table or a list. Experiments using coins, dice, cards and so on, can enhance understanding of the distinction between
experimental (relative frequency) and theoretical probability. Expected number of occurrences. Example: If there are 128
students in a class and the probability of being absent is 0.1, the expected number of absent students is 12.8.
SL 4.6 Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities. Combined
events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B). Mutually exclusive events: P (A ∩ B) = 0. The non-exclusivity of “or”. Conditional
probability: P(A|B) = P(A ∩ B)/P(B) . An alternate form of this is: P(A ∩ B) = P(B)P(A|B). Problems can be solved with the aid of a
Venn diagram, tree diagram, sample space diagram or table of outcomes without explicit use of formulae. Probabilities with and
without replacement. Independent events: P(A ∩ B) = P(A)P(B).
20
Core topics:
SL 4.1 Concepts of population, sample, random sample, discrete and continuous data. This is designed to cover the key questions
Sampling and
that students should ask when they see a data set/analysis. Reliability of data sources and bias in sampling. Dealing with missing
Data
data, errors in the recording of data. Interpretation of outliers. Outlier is defined as a data item which is more than 1.5 ×
interquartile range (IQR) from the nearest quartile. Awareness that, in context, some outliers are a valid
part of the sample but some outlying data items may be an error in the sample. Sampling techniques and their effectiveness.
Simple random, convenience, systematic, quota and stratified sampling methods.
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1/11
1/12
21
Core topics:
Statistics
22
Discrete Random
Variables
23
The Normal
Distribution
24
Estimation and
Confidence
Intervals
AHL 4.12 Design of valid data collection methods, such as surveys and questionnaires. Selecting relevant variables from many
variables. Choosing relevant and appropriate data to analyse. Biased and unbiased, personal, unstructured and structured (with
consistent answer choices), and precise questioning.
SL 4.2 Presentation of data (discrete and continuous): frequency distributions (tables). Class intervals will be given as inequalities,
without gaps. Histograms. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range
and interquartile range (IQR). Frequency histograms with equal class intervals. Not required: Frequency density histograms.
Production and understanding of box and whisker diagrams. Use of box and whisker diagrams to compare two distributions, using
symmetry, median, interquartile range or range. Outliers should be indicated with a cross. Determining whether the data may be
normally distributed by consideration of the symmetry of the box and whiskers.
SL 4.3 Measures of central tendency (mean, median and mode). Estimation of mean from grouped data. Calculation of mean
using formula and technology. Students should use mid-interval values to estimate the mean of grouped data. Modal class. For
equal class intervals only. Measures of dispersion (interquartile range, standard deviation and variance). Calculation of standard
deviation and variance of the data set using only technology, however hand calculations may enhance understanding. Variance is
the square of the standard deviation. Effect of constant changes on the original data. Examples: If three is subtracted from the
data items, then the mean is decreased by three, but the standard deviation is unchanged. If all the data items are doubled, the
mean is doubled and the standard deviation is also doubled. Quartiles of discrete data. Using technology. Awareness that
different methods for finding quartiles exist and therefore the values obtained using technology and by hand may differ.
SL 4.7 Concept of discrete random variables and their probability distributions. Expected value (mean), E(X) for discrete data.
Applications. E(X) = 0 indicates a fair game where X represents the gain of a player.
SL 4.8 Binomial distribution. Mean and variance of the binomial distribution. Situations where the binomial distribution is an
appropriate model. In examinations, binomial probabilities should be found using available technology (formula not required, and
questions asking to find, for example, unknown p with P(X=1) given not examined).
AHL 4.17 Poisson distribution (formula not required), its mean and variance. Sum of two independent Poisson distributions has a
Poisson distribution. Situations in which it is appropriate to use a Poisson distribution as a model: 1. Events are independent 2.
Events occur at a uniform average rate (during the period of interest). Given a context, students should be able to select between
the normal, the binomial and the Poisson distributions, recognizing where a particular distribution is appropriate.
SL 4.9 The normal distribution and curve. Properties of the normal distribution. Diagrammatic representation. Awareness of the
natural occurrence of the normal distribution. Students should be aware that approximately 68% of the data lies between μ ± σ,
95% lies between μ ± 2σ and 99.7% of the data lies between μ ± 3σ. Normal probability calculations. Probabilities and values of
the variable must be found using technology. Inverse normal calculations. For inverse normal calculations mean and standard
deviation will be given. This does not involve transformation to the standardized normal variable z.
AHL 4.14 Linear transformation of a single random variable. Var X is the variance of the random variable X. Variance formula will
not be required in examinations. E(aX + b) = aE(X) + b. Var (aX + b) = a2Var(X). Expected value of linear combinations of n random
𝑥
2
variables. Variance of linear combinations of n independent random variables. x¯ as an unbiased estimate of μ. 𝑥̅ = ∑𝑛𝑖=1 𝑛𝑖. 𝑠𝑛−1
𝑛
2
as an unbiased estimate of 𝜎 2 . 𝑠𝑛−1
= 𝑛−1 𝑠𝑛2 = ∑𝑘𝑖=1
will not be examined, but may help understanding.
𝑓𝑖 (𝑥−𝑥̅ )2
,
𝑛−1
2
where 𝑛 = ∑𝑛𝑖=1 𝑓𝑖 . Demonstration that E(𝑋̅) = μ and E(𝑠𝑛−1
) = σ2
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2/27
2/28
2/29
AHL 4.15 A linear combination of n independent normal random variables is normally distributed. In particular, 𝑋~𝑁(𝜇, 𝜎 2 ) ⇒
𝜎2
𝑋̅~𝑁 (𝜇, ). Central limit theorem. In general, 𝑋̅ approaches normality for large n, how large depends upon the distribution from
𝑛
which the sample is taken. In examinations, n > 30 will be considered sufficient.
AHL 4.16 Confidence intervals for the mean of a normal population. Students should be able to interpret the meaning of their
results in context. Use of the normal distribution when σ is known and the t-distribution when σ is unknown, regardless of sample
size.
25
Hypothesis
SL 4.11 Formulation of null and alternative hypotheses, H0 and H1. Significance levels. p -values. Students should express H0 and H1 2/30
Testing
as an equation or inequality, or in words as appropriate. The t -test. Use of the p -value to compare the means of two populations.
Using one-tailed and two-tailed tests. In examinations calculations will be made using technology. Students will be asked to
interpret the results of a test. Students should know that the underlying distribution of the variables must be normal for the t-test
to be applied. In examinations, students should assume that variance of the two groups is equal and therefore the pooled twosample t -test should be used.
AHL 4.18 Critical values and critical regions. Students will not be expected to calculate critical regions for t-tests. Test for
population mean for normal distribution. Use of the normal distribution when σ is known and the t-distribution when σ is
unknown, regardless of sample size. The case of matched pairs is to be treated as an example of a single sample technique. Test
for proportion using binomial distribution. Test for population mean using Poisson distribution. Poisson and binomial tests will be
one-tailed only. Use of technology to test the hypothesis that the population product moment correlation coefficient (ρ) is 0 for
bivariate normal distributions. In examinations the data will be given. Type I and II errors including calculations of their
probabilities. Applied to normal with known variance, Poisson and binomial distributions.
For discrete random variables, hypothesis tests and critical regions will only be required for one-tailed tests. The critical region
will maximize the probability of a Type I error while keeping it less than the stated significance level.
26
Chi Squared
SL 4.11 Expected and observed frequencies. The χ2 test for independence: contingency tables, degrees of freedom, critical value.
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Hypothesis Tests
The χ2 goodness of fit test. In examinations: the maximum number of rows or columns in a contingency table will be 4, the
degrees of freedom will always be greater than one. The χ2 critical value will be given if appropriate, students will be expected to
use technology to find a p -value and the χ2 statistic, only questions on upper tail tests with commonly-used significance levels
(1%, 5%, 10%) will be set, students will be expected to either compare a p-value to the given significance level or compare the χ2
statistic to a given critical value, expected frequencies will be greater than 5. Hand calculations of the expected values or the χ2
statistic may enhance understanding. If using χ2 tests in the IA, students should be aware of the limitations of the test for
expected frequencies of 5 or less.
AHL 4.12 Categorizing numerical data in a χ2 table and justifying the choice of categorisation. Choosing an appropriate number of
degrees of freedom when estimating parameters from data when carrying out the χ2 goodness of fit test. Appropriate categories
should be chosen with expected frequencies greater than 5. Definition of reliability and validity. Reliability tests. Validity tests.
Students should understand the difference between reliability and validity and be familiar with the following methods: Reliability:
Test-retest, parallel forms. Validity: Content, criterion-related.
27
Recap, review and assessment
Unit 4: Internal Assessment
6th May
IA aim and rationale due
3rd June
IA outline due
13th – 15th June
End of year exams
Year 2
Week
Content
1
IA
Unit 1: Trigonometry
2
Core topics:
Measurement
and RightAngled Triangle
Trigonometry
3
Core topics: The
Unit Circle and
Radian Measure
4
Core topics:
Non-Right
Angled Triangle
Trigonometry
Core topics:
Trigonometric
Functions
5
Syllabus Reference
Book/
Chapter
SL 3.1 The distance between two points in three-dimensional space, and their midpoint. Volume and surface area of threedimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle
between two intersecting lines or between a line and a plane. In problems related to these topics, students should be able to
identify relevant right-angled triangles in three-dimensional objects and use them to find unknown lengths and angles.
SL 3.2 Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
SL 3.3 Applications of right angled trigonometry, including Pythagoras’ theorem. Angles of elevation and depression.
Construction of labelled diagrams from written statements. Contexts may include use of bearings.
SL 3.4 The circle: length of an arc; area of a sector.
IA draft for peer feedback due
AHL 3.7 The definition of a radian and conversion between degrees and radians. Using radians to calculate area of sector, length
of arc. Radian measure may be expressed as exact multiples of π, or decimals.
AHL 3.8 The definitions of cosθ and sinθ in terms of the unit circle. The Pythagorean identity: cos2θ + sin2θ = 1, definition of tanθ
as sinθ/cosθ. Extension of the sine rule to the ambiguous case. Students should understand how the graphs of f(x) = sinx and f(x)
= cosx can be constructed from the unit circle. Knowledge of exact values of cosθ, sinθ, and tanθ will not be assessed on
examinations, but may aid student understanding of trigonometric functions.
SL 3.2 The sine rule, the cosine rule, trigonometric formula for the area of a triangle. This section does not include the ambiguous
case of the sine rule.
SL 3.3 Applications of non-right angled trigonometry.
1/6 and
7
SL 2.5 Modelling with sinusoidal models. Students will not be expected to translate between sinx and cosx, and will only be
required to predict or find amplitude (|a|), period (360∘/b), or equation of the principal axis (y = d).
AHL 2.9 Sinusoidal models: f x = asin(b(x – c)) + d. Radian measure should be assumed unless otherwise indicated by the use of
the degree symbol, for example with f (x) = sinx°. In radians, period is 2π/b. Students should be aware that a horizontal
translation of c can be referred to as a phase shift.
AHL 3.8 Graphical methods of solving trigonometric equations in a finite interval.
6
Recap, review and assessment
Unit 2: Vectors, Complex Numbers and Matrices
1/8
1/9 and
10
1/17
7
Vectors
AHL 3.10 Concept of a vector and a scalar. Representation of vectors using directed line segments. Unit vectors; base vectors i, j,
𝑣1
k. Components of a vector; column representation; 𝒗 = (𝑣2 ) = 𝑣1 𝒊 + 𝑣2 𝒋 + 𝑣3 𝒌. The zero vector 0, the vector −v. Use
𝑣3
algebraic and geometric approaches to calculate the sum and difference of two vectors, multiplication by a scalar, kv (parallel
⃗⃗⃗⃗⃗ =
vectors), magnitude of a vector |v| from components. The resultant as the sum of two or more vectors. Position vectors 𝑂𝐴
𝑎. Rescaling and normalizing vectors. v/|v|, the unit normal vector. Example: Find the velocity of a particle with speed
7ms-1 in the direction 3i + 4j.
AHL 3.13 Definition and calculation of the scalar product of two vectors. The angle between two vectors; the acute angle
between two lines. Calculate the angle between two vectors using v · w = |v||w|cosθ, where θ is the angle between two nonzero vectors v and w, and ascertain whether the vectors are perpendicular (v · w = 0). Definition and calculation of the vector
product of two vectors. v × w = |v| |w| sinθn, where θ is the angle between v and w and n is the unit normal vector whose
direction is given by the right-hand screw rule. Geometric interpretation of | v × w |. Use of | v × w | to find the area of a
parallelogram (and hence a triangle). Components of vectors. The component of vector a acting in the direction of
𝑎∙𝑏
vector b is |𝑏| = |𝑎| cos 𝜃. The component of a vector a acting perpendicular to vector b, in the plane formed by the two vectors,
is
8
Vector
Applications
9
Complex
Numbers
10
Matrices
|𝑎×𝑏|
|𝑏|
2/9
= |𝑎| sin 𝜃.
AHL 3.11 Vector equation of a line in two and three dimensions: r = a + λb, where b is a direction vector of the line. Convert to
parametric form: x = x0 + λl, y = y0 + λm, z = z0 + λn.
AHL 3.12 Vector applications to kinematics. Modelling linear motion with constant velocity in two and three dimensions. Finding
positions, intersections, describing paths, finding times and distances when two objects are closest to each other. r = r0 + vt.
Relative position of B from A is ⃗⃗⃗⃗⃗
𝐴𝐵. Motion with variable velocity in two dimensions. Projectile motion and circular motion are
special cases. f(t – a) to indicate a time-shift of a.
AHL 1.12 Complex numbers: the number i such that i2 = − 1. Cartesian form: z = a + bi; the terms real part, imaginary part,
conjugate, modulus and argument. Calculate sums, differences, products, quotients, by hand and with technology. Calculating
powers of complex numbers, in Cartesian form, with technology. The complex plane. Use and draw Argand diagrams. Complex
numbers as solutions to quadratic equations of the form ax2 + bx + c = 0, a ≠ 0, with real coefficients where b2 − 4ac < 0.
Quadratic formula and the link with the graph of f(x) = ax2 + bx + c.
AHL 1.13 Modulus–argument (polar) form: z = r cosθ + isinθ = rcisθ. Exponential form: z = reiθ. Exponential form is sometimes
called the Euler form. Conversion between Cartesian, polar and exponential forms, by hand and with technology. Calculate
products, quotients and integer powers in polar or exponential forms. In examinations students will not be required to find the
roots of complex numbers. Adding sinusoidal functions with the same frequencies but different phase shift angles. Phase shift
and voltage in circuits as complex quantities. Example: Two AC voltage sources are connected in a circuit. If V1 = 10 cos (40t) and
V2 = 20 cos (40t + 10) find an expression for the total voltage in the form V = A cos (40t + B). Geometric interpretation of complex
numbers. Addition and subtraction of complex numbers can be represented as vector addition and subtraction. Multiplication of
complex numbers can be represented as a rotation and a stretch in the Argand diagram.
AHL 1.14 Definition of a matrix: the terms element, row, column and order for m × n matrices. Algebra of matrices: equality;
addition; subtraction; multiplication by a scalar for m × n matrices. Including use of technology. Multiplication of matrices.
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Properties of matrix multiplication: associativity, distributivity and non-commutativity. Multiplying matrices to solve practical
problems. Identity and zero matrices. Determinants and inverses of n × n matrices with technology, and by hand for 2 × 2
matrices. Students should be familiar with the notation I and 0. Awareness that a system of linear equations can be written in
the form Ax = b. In examinations A will always be an invertible matrix, except when solving for eigenvectors. Solution of the
systems of equations using inverse matrix. Model and solve real-life problems including: Coding and decoding messages, Solving
systems of equations.
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Eigenvalues and AHL 1.15 Eigenvalues and eigenvectors. Characteristic polynomial of 2 × 2 matrices. Diagonalization of 2 × 2 matrices (restricted
Eigenvectors
to the case where there are distinct real eigenvalues). Students will only be expected to perform calculations by hand and with
technology for 2 × 2 matrices. Applications to powers of 2 × 2 matrices. Applications, for example movement of population
between two towns, predator/prey models. Mn = PDnP−1, where P is a matrix of eigenvectors, and D is a diagonal matrix of
eigenvalues.
AHL 4.19 Transition matrices. Powers of transition matrices. In general, the column state matrix (sn) after n transitions is given by
sn = Tns0, where T is the transition matrix, with Tij representing the probability of moving from state j to state i, and s0 is the initial
state matrix. Use of transition diagrams to represent transitions in discrete dynamical systems. Regular Markov chains. Initial
state probability matrices. Calculation of steady state and long-term probabilities by repeated multiplication of the transition
matrix or by solving a system of linear equations. Examination questions will state when exact solutions obtained from solving
equations are required. Awareness that the solution is the eigenvector corresponding to the eigenvalue equal to 1 and whose
entries have been scaled to sum to 1.
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Affine
AHL 3.9 Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches,
Transformations enlargements, translations and rotations. Matrix transformations of the form: (𝑎 𝑏 ) (𝑥 ) + ( 𝑒 ). Compositions of the above
𝑓
𝑐 𝑑 𝑦
transformations. Iterative techniques to generate fractals. Geometric interpretation of the determinant of a transformation
matrix. Area of image = | detA | × area of object.
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Recap, review and assessment
Unit 3: Graph Theory and Voronoi Diagrams
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Graph Theory
AHL 3.14 Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges. Degree of a vertex. Students should be able to
represent real-world structures (circuits, maps, etc) as graphs (weighted and unweighted). Simple graphs; complete graphs;
weighted graphs. Knowledge of the terms connected and strongly connected. Directed graphs; in degree and out degree of a
directed graph. Subgraphs; trees.
AHL 3.15 Adjacency matrices. Walks. Number of k -length walks (or less than k -length walks) between two vertices. Given an
adjacency matrix A, the (i, j)th entry of Ak gives the number of k length walks from i to j. Situations may not be symmetric.
Weighted adjacency tables. Construction of the transition matrix for a strongly connected, undirected or directed graph. Weights
could be costs, distances, lengths of time
for example. Consideration of simple graphs, including the Google PageRank algorithm as an example of this.
AHL 3.16 Tree and cycle algorithms with undirected graphs. Walks, trails, paths, circuits, cycles. Eulerian trails and circuits.
Hamiltonian paths and cycles. Minimum spanning tree (MST) graph algorithms: Kruskal’s and Prim’s algorithms for finding
minimum spanning trees. Determine whether an Eulerian trail or circuit exists. Use of matrix method for Prim’s algorithm.
Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four
odd vertices, going along each edge at least once. Students should be able to explain why the algorithm for constructing the
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Chinese postman problem works, apply the algorithm and justify their choice of algorithm. Travelling salesman problem to
determine the Hamiltonian cycle of least weight in a weighted complete graph. Nearest neighbour algorithm for determining an
upper bound for the travelling salesman problem. Deleted vertex algorithm for determining a lower bound for the travelling
salesman problem. Practical problems should be converted to the classical problem by completion of a table of least distances
where necessary.
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Voronoi
SL 3.6 Voronoi diagrams: sites, vertices, edges, cells. Addition of a site to an existing Voronoi diagram. Nearest neighbour
Diagrams
interpolation. Applications of the “toxic waste dump” problem. In examinations, coordinates of sites for calculating the
perpendicular bisector equations will be given. Students will not be required to construct perpendicular bisectors. Questions may
include finding the equation of a boundary, identifying the site closest to a given point, or calculating the area of a region. All
points within a cell can be estimated to have the same value (e.g. rainfall) as the value of the site. In examinations, the solution
point will always be at an intersection of three edges. Contexts: Urban planning, spread of diseases, ecology, meteorology,
resource management.
IA first draft for formal teacher feedback due
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Recap, review and assessment
Unit 4: Calculus
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Introduction to
SL 5.1 Introduction to the concept of a limit. Estimation of the value of a limit from a table or graph. Not required: Formal
Differential
analytic methods of calculating limits. Derivative interpreted as gradient function and as rate of change. Forms of notation:
Calculus
dy/dx, f ′(x), dV/dr or ds/dt for the first derivative. Informal understanding of the gradient of a curve as a limit.
SL 5.2 Increasing and decreasing functions. Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0. Identifying intervals on which
functions are increasing ( f ′(x) > 0) or decreasing ( f ′(x) < 0).
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Rules of
SL 5.3 Derivative of f (x) = axn is f ′(x) = anxn − 1, n ∈ ℤ The derivative of functions of the form f(x) = axn + bxn − 1 +... where all
Differentiation
exponents are integers.
AHL 5.9 The derivatives of sin x, cos x, tan x, ex, ln x, xn where n ∈ ℚ. The chain rule, product rule and quotient rules. Related
rates of change.
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Mock Exams (4th - 11th Feb)
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Properties of
SL 5.4 Tangents and normals at a given point, and their equations. Use of both analytic approaches and technology.
Curves
SL 5.6 Values of x where the gradient of a curve is zero. Solution of f ′(x) = 0. Local maximum and minimum points. Students
should be able to use technology to generate f ′(x) given f (x), and find the solutions of f ′(x) = 0. Awareness that the local
maximum/minimum will not necessarily be the greatest/least value of the function in the given domain.
AHL 5.9 The second derivative. Both forms of notation, d2y/dx2 and f ′′(x) for the second derivative. Use of second derivative test
to distinguish between a maximum and a minimum point. Awareness that a point of inflexion is a point at which the concavity
changes and interpretation of this in context. Use of the terms “concave-up” for f′′(x) > 0, and “concave-down” for f′′(x) < 0.
IA final due
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Applications of
SL 5.7 Optimisation problems in context. Examples: Maximizing profit, minimizing cost, maximizing volume for a given surface
Differentiation
area.
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Introduction to
SL 5.5 Introduction to integration as anti-differentiation of functions of the form f (x) = axn + bxn − 1 + ...., where n ∈ ℤ, n ≠ − 1.
Integration
Students should be aware of the link between antiderivatives, definite integrals and area. Anti-differentiation with a boundary
condition to determine the constant term. Example: If dy/dx = 3x2 + x and y = 10 when x = 1, then y = x3 +12x2 + 8.5.
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SL 5.8 Approximating areas using the trapezoidal rule. Given a table of data or a function, make an estimate
for the value of an area using the trapezoidal rule, with intervals of equal width.
Techniques for
AHL 5.11 Definite and indefinite integration of xn where n ∈ ℚ, including n = − 1 , sin x, cos x, 1/cos2x and ex. Integration by
Integration
inspection, or substitution of the form ∫ f (g(x))g′(x)dx. Examples:∫sin(2x + 5)dx, ∫ 1/(3x + 2)dx, ∫4xsinx2 dx, ∫sinx/cosx dx.
Definite
SL 5.5 Definite integrals using technology. Area of a region enclosed by a curve y = f (x) and the x-axis, where f (x) > 0.
Integrals
Students are expected to first write a correct expression before calculating the area. The use of dynamic geometry or graphing
software is encouraged in the development of this concept.
AHL 5.12 Area of the region enclosed by a curve and the x or y-axes in a given interval. Including negative integrals. Volumes of
revolution about the x- axis or y- axis.
Kinematics
AHL 5.13 Kinematic problems involving displacement s, velocity v and acceleration a. v = ds/dt ; a = dv/dt = d2s/dt2 = vdv/ds.
Displacement as the integral of velocity. Total distance travelled as the integral of the magnitude of the velocity. Speed is the
magnitude of velocity. Use of 𝑥̇ = dx/dt and 𝑥̈ = d2x/dt2.
Differential
AHL 5.14 Setting up a model/differential equation from a context. Example: The growth of an algae G, at time t, is proportional
Equations
to √𝐺. Solving by separation of variables. Example: An exponential model as a solution of dy/dx = ky. The term “general
solution”.
AHL 5.15 Slope fields and their diagrams. Students will be required to use and interpret slope fields.
Coupled
AHL 5.16 Euler’s method for finding the approximate solution to first order differential equations. Numerical solution of dy/dx = f
Differential
(x, y). Spreadsheets should be used to find approximate solutions to differential equations. In examinations, values will be
Equations
generated using permissible technology. Numerical solution of the coupled system dx/dt = f1(x, y, t) and dy/dt = f2(x, y, t).
Contexts could include predator-prey models.
AHL 5.17 Phase portrait for the solutions of coupled differential equations of the form: dx/dt = ax + by and dy/dt = cx + dy.
Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues. Sketching trajectories and using phase
portraits to identify key features such as equilibrium points, stable populations and saddle points. Systems will have distinct,
non-zero, eigenvalues. If the eigenvalues are: Positive or complex with positive real part, all solutions move away from the origin,
Negative or complex with negative real part, all solutions move towards the origin, Complex, the solutions form a spiral,
Imaginary, the solutions form a circle or ellipse, Real with different signs (one positive, one negative) the origin is a saddle point.
Calculation of exact solutions is only required for the case of real distinct eigenvalues.
AHL 5.18 Solutions of d2x/dt2 = f (x, dx/dt, t) by Euler’s method and by finding exact solutions. Write as coupled first order
equations dx/dt = y and dy/dt = f (x, y, t). Solutions of d2x/dt2 + adx/dt + bx = 0, can also be investigated using the phase portrait
method in AHL 5.17. Understanding the occurrence of simple second order differential equations in physical phenomena would
aid understanding but in examinations the equation will be given.
Recap, review and assessment
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